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A representation of the hazard rate of elapsed time in
macaque area LIP
Peter Janssen1,2 & Michael N Shadlen1
The capacity to anticipate the timing of environmental cues allows us to allocate sensory resources at the right time and prepare
actions. Such anticipation requires knowledge of elapsed time and of the probability that an event will occur. Here we show
that neurons in the parietal cortex represent the probability, as a function of time, that a salient event is likely to occur. Rhesus
monkeys were trained to make eye movements to peripheral targets after a light dimmed. Within a block of trials, the ‘go’ times
were drawn from either a bimodal or unimodal distribution of random numbers. Neurons in the lateral intraparietal area showed
anticipatory activity that revealed an internal representation of both elapsed time and the probability that the ‘go’ signal was
about to occur (termed the hazard rate). The results indicate that the parietal cortex contains circuitry for representing the time
structure of environmental cues over a range of seconds.
Humans and animals rely on a sense of elapsed time to plan actions,
anticipate salient events, infer causal regularities and learn associations1.
Yet little is known about the neural mechanisms that underlie the
encoding and use of elapsed time2,3. Traditionally, the focus has been
on the cerebellum and basal ganglia, but several recent studies suggest that the association areas of the neocortex may play an important role4–6. Neurons in association areas perform computations that
span gaps in time between sensation and action in order to mediate
working memory7, motor planning8,9 and decision making10–12. These
processes depend on a representation of time to infer temporal order,
plan sequences of actions and control the tradeoff between speed and
accuracy. In general, processes not controlled by immediate external
events may still need to know when information is useful.
Behavioral performance is enhanced if subjects can anticipate the
point in time that a stimulus is likely to appear or an instruction is likely
to occur4,13–15. To anticipate the timing of behaviorally relevant events,
the brain must represent the passage of time and use this representation
to estimate the probability that an event is likely to occur, given that it
has not occurred already. This computation is termed a hazard rate14.
Although several brain regions have been shown to encode the
probability that an action will ensue12,16,17, little is known about how
such probabilities are represented by the brain when they change as
a function of time. Neurons in various brain areas have occasionally
shown climbing activity in the interval preceding a test stimulus or ‘go’
instruction, an effect that has been interpreted as a neural correlate
of anticipation18–23. A recent study showed that neurons in the lateral intraparietal area (LIP) undergo time-dependent changes in their
responses when monkeys make decisions about the duration of a time
interval5. Because LIP neurons show sustained changes in firing rate
during delayed eye movement tasks9,24–26, we hypothesized that this
persistent activity might encode the hazard rate used to anticipate the
time of a pending ‘go’ signal.
RESULTS
Two rhesus monkeys were trained to make eye movements to a
peripheral target (Fig. 1a). The monkeys were required to hold steady
fixation until the fixation spot had dimmed and were then rewarded for
initiating an accurate eye movement as soon as possible (see Methods).
The waiting time between target onset and the ‘go’ signal (that is, the ‘go’
time) was a random variable whose probability distribution was fixed
throughout a block of trials. We used two time schedules in alternating
blocks of trials. In the bimodal time schedule (Fig. 1b, top row left),
the ‘go’ signal could come either early or late, but not in the interval
between 0.75 and 1.75 s, whereas in the unimodal time schedule (Fig 1b,
top row right), the ‘go’ times were distributed between 0.5 and 2 s.
We reasoned that exposure to these schedules might allow the
monkeys to anticipate the arrival of the ‘go’ signal. The time course of
such anticipation is formalized by the hazard rate. Because the brain
cannot estimate elapsed time precisely1, the mathematical functions
are replaced by blurred versions, which we term subjective hazard rates
or anticipation functions (see Methods). There are clear differences
between the anticipation functions associated with the unimodal and
bimodal schedules (Fig. 1b, bottom). When ‘go’ times are drawn from
the unimodal distribution, the anticipation function mainly increases
with waiting time. In contrast, when ‘go’ times are drawn from the
bimodal distribution, the anticipation is triphasic: it rises, falls and
1Howard
Hughes Medical Institute, National Primate Research Center and Department of Physiology and Biophysics, University of Washington, Box 357290,
Seattle, Washington 98195. 2Present address: Laboratorium voor Neuro-en Psychofysiologie, KU Leuven Medical School, Herestraat 49, B-3000 Leuven, Belgium.
Correspondence should be addressed to M.N.S. ([email protected]).
Published online 16 January 2005; doi:10.1038/nn1386
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Figure 1 Methods. (a) Delayed eye movement task. The monkey made
eye movements to the red target as soon as the fixation point dimmed.
The target appeared in different locations on each trial. Here we report
trials in which the target appeared in the response field of the LIP neuron
we recorded. A bracket demarcates the random waiting time between
target onset and ‘go’ signal. (b) Probability distributions, hazard rates and
subjective hazard rates. The top row illustrates the probability distributions
for drawing ‘go’ times under the two schedules used in the experiments:
bimodal (left) and unimodal (right). The middle row shows the hazard
rates for each of these time schedules (equation (3)). The bottom row shows
the subjective hazard rates or anticipation functions for each of these
time schedules (Ab(t) and Au(t), φ = 0.26, equation (4)). The functions
employ a common scale, which is normalized to the peak of the Au(t).
(c) Representative magnetic resonance image from monkey J. Neurons in
both monkeys were recorded from the posterior portion of LIPv27, outlined
by the arrows; ips, intraparietal sulcus.
P < 0.001). In ‘bimodal schedule’ blocks, reaction time decreased and
rose again for ‘go’ times that happened to be drawn from the first mode
and was fastest for ‘go’ times that were drawn from the second mode.
This triphasic pattern was more striking for monkey J (Fig. 2a), but it
was evident in both monkeys, as shown below. The reaction times from
both monkeys clearly decreased between the early and later modes (for
monkey J, mean reaction time was 268 ± 1.2 and 237 ± 0.8 ms in epochs
within ±350 ms of the first and second modes, respectively; for monkey
H, the corresponding means were 295 ± 0.7 and 259 ± 1).
The monkeys' reaction times were inversely related to the anticipation
functions associated with the schedules of random 'go' times. For
either schedule, the data were well fit by a weighted sum of the two
anticipation functions delayed by 56 ms (Methods, equation (5)). These
fits furnish estimates of the number of milliseconds that reaction time
is reduced per unit change in anticipation (Table 1). Under the bimodal
schedule, the fit to the data from monkey J (Fig. 2a) is dominated by the
bimodal anticipation function, whereas under the unimodal schedule,
the fit is dominated by the unimodal anticipation function (Table 1).
For monkey H, the unimodal anticipation function explains most of
the decline in reaction time under both schedules. Nevertheless, the
bimodal anticipation function has a significant role under the bimodal
schedule of ‘go’ times (P < 0.02); (Table 1).
The influence of the bimodal anticipation function can be understood
more directly by examining the partial correlation between the reaction
times from individual trials and the anticipation function appropriate
for the schedule. Because both anticipation functions can affect the
pattern of reaction times under either schedule, we performed a partial
(conditional) correlation to factor out the influence of the potentially
confounding anticipation function. Under the bimodal schedule, the
partial correlation between reaction time and the bimodal anticipation function, rRTb ,Ab|Au, was –0.36 for monkey J (P < 0.001, Fisher z).
Under the unimodal schedule, there was a significant inverse correlation between reaction time and the unimodal anticipation function
(rRTu,Au|Ab = –0.24; P < 0.001). A similar inverse correlation was present
rises again. The scale of these anticipation functions was set from 0 to
1 to facilitate interpretation of the behavioral and physiological data,
as described below.
Anticipation of the ‘go’ cue affects saccade reaction time
Both monkeys learned to anticipate the time of the ‘go’ signal, as
evidenced by their reaction times. Figure 2 shows running means of
reaction times measured under the unimodal (purple) and bimodal
(red) schedules. Each point shows the mean of 51 observations. In both
types of blocks, reaction time showed a clear dependency on the amount
of time that elapsed before the ‘go’ signal. In ‘unimodal schedule’ blocks,
reaction time decreased with longer waiting times (regression analysis,
Table 1 Weighting coefficients for the unimodal and bimodal anticipation functions fit to reaction time data in Figure 2, using equation (5)
Weights of anticipation functions fit to reaction times (ms change in reaction time per unit anticipation)
Monkey J
Distribution of ‘go’ times
Monkey H
R2
R2
wu
wb
Bimodal schedule
–42.2 ± 1.4
–69.8 ± 4.2
0.96 (0.29)
–40.0 ± 2.0
–9.7 ± 3.9
0.95 (0.39)
Unimodal schedule
–17.9 ± 2.2
–9.7 ± 6.9*
0.77 (0.06)
–32.9 ± 2.6
–22.1 ± 9.1
0.91 (0.11)
wu
wb
The coefficients multiply the anticipation functions, Au(t) and Ab(t). One unit of anticipation is the range of Au(t) shown in Figure 1. Both anticipation functions affected reaction time inversely
under both schedules (P < 0.02, except for asterisk). R2 describes the fraction of the running mean variance explained by the fit; parenthetical values describe the fraction of variance for
individual trials.
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Figure 2 Reaction time (RT) is modulated by anticipation of the ‘go’ cue.
Eye movement RTs are plotted as a function of the waiting time between
target onset and the ‘go’ signal (the ‘go’ time), during the bimodal (red) and
the unimodal (purple) time schedules. Points represent running means of
RT from 51 consecutive ‘go’ times. Black curves are fits to the data using a
weighted sum of the anticipation functions (equation (5); weights are in Table
1). The dashed red curve is the bimodal anticipation function sampled at the
‘go’ times that occur under the bimodal schedule. The dashed purple curve
shows the unimodal anticipation function sampled at the ‘go’ times under the
unimodal schedule. Note that RTs are only measured when there is a finite
probability that a ‘go’ cue could occur. (a) Data from monkey J (n = 2,322
unimodal trials and n = 1,876 bimodal trials). (b) Data from monkey H (n =
1,480 unimodal trials and n = 1,456 bimodal trials).
in monkey H (rRTb ,Ab|Au = –0.09; rRTu ,Au|Ab = –0.32; P < 0.001 for both
values). These behavioral observations imply that the brain is capable
of representing both the passage of time and the time-dependent probability that the ‘go’ signal is about to occur.
Anticipation of the ‘go’ cue affects LIP neural activity
We recorded 70 LIP neurons, screening them using an eye movement
task in which monkeys made saccadic eye movements to the
remembered location of a briefly flashed target. We studied only
those neurons that showed strong, spatially selective activity during
the memory period between the flashed eye movement target and
the ‘go’ instruction (see Methods). These neurons were commonly
encountered in the ventral portion of LIP27 (Fig. 1c). We then tested
these neurons using the delayed saccadic eye movement task with a
bimodal or unimodal distribution of random ‘go’ times. In half the
trials, the saccade target was in the response field of the LIP neuron,
giving rise to a persistent elevation in the neural activity. Our focus is
on the modulation of this persistent activity while the monkey awaited
the ‘go’ instruction.
Many neurons in area LIP altered their firing rates as a function of
time, in accordance with the changing anticipation of the ‘go’ signal.
Figure 3a shows responses of a representative neuron recorded in
blocks of trials using either the bimodal or the unimodal schedule of
‘go’ times. Each curve represents averaged spike rates measured during
Figure 3 LIP responses represent anticipation as a
function of time. (a) Single-neuron example. Top:
anticipation functions for the unimodal (purple) and
the bimodal (red) time schedule. Bottom: average
neural activity recorded during waiting period for the
‘go’ signal under the bimodal (red) and the unimodal
(purple) time schedule. Shading, s.e.m.; black curves,
fits to equation (5). (b) Population responses from
blocks using bimodally and unimodally distributed
‘go’ times. Averaged normalized responses (± s.e.m.)
plotted as a function of waiting time for monkey J
(n = 39) and monkey H (n = 31). Spike rates from
each neuron were normalized to the mean activity
320–1,360 ms after target onset using all trials in
both schedules. Black curves, fits to equation (5);
red, bimodal schedule; purple, unimodal schedule.
(c) Effect of schedule on the temporal pattern of the
response from single neurons. Average response from a
block using either bimodal or unimodal schedule of ‘go’
times was described as a weighted sum of anticipation
functions, Au(t) and Ab(t). Upper scatter plot compares
amount of response modulation attributed to Ab(t) in
unimodal and bimodal testing blocks. The bimodal
anticipation function explained more of the response
modulation during the block of trials using bimodally
distributed ‘go’ times (mean wb was +14.1 and
+0.8 during the bimodal and unimodal schedules,
respectively). Differences are summarized by the
frequency histogram. Lower scatter plot compares
amount of response modulation attributed to Au(t) in
unimodal and bimodal testing blocks. The unimodal
anticipation function explained more of the response
modulation during the block of trials using unimodally
distributed ‘go’ times (mean wu was +4.8 and –1.1
during unimodal and bimodal schedules, respectively).
Differences are summarized by the frequency
histogram. Error bars, s.d. of parameter estimates.
Shaded histograms indicate significant cases, P <
0.01. Green symbols indicate example neuron in a.
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Table 2 Weighting coefficients for the unimodal and bimodal anticipation
functions fit to response averages in Figure 3b, using equation (5)
Weights of anticipation functions fit to LIP response
(Percentage change in firing rate per unit anticipation)
Monkey J
© 2005 Nature Publishing Group http://www.nature.com/natureneuroscience
Distribution of ‘go’ times
wu
Bimodal schedule
–19 ± 1
Unimodal schedule
+25 ± 2
Monkey H
wb
wu
+51.4 ± 1.9 –7.1 ± 1.8
+26 ± 2.3
+11 ± 1.4
wb
+39 ± 2.7
+22 ± 2.3
The coefficients multiply the anticipation functions, Au(t) and Ab(t). Units for the population
responses are percentage change in firing rate (relative to the average delay period activity) per
unit change in the anticipation function. One unit of anticipation is the range of Au(t) shown
in Figure 1b. Schedule affected the weights in the appropriate direction (compare weights in
each column; P < 10–5 for all comparisons).
the waiting period from the onset of the eye movement target at the
cell’s preferred location until the dimming of the fixation point.
When the neuron was studied in a bimodal ‘go’ time schedule, the
firing rate showed an early peak in activity at ∼0.4 s after target onset.
Then, if the 'go' signal did not arrive, the activity declined by 41 spikes
s–1 over the following 0.4 s. Then, from ∼0.8 s, the response increased
gradually by 24 spikes s–1 as the monkey awaited a ‘go’ signal drawn
from the later mode of the bimodal distribution. When the neuron was
studied with a unimodal ‘go’ time schedule, the early rise in activity was
conspicuously absent. After the onset response, the firing rate declined
by 9 spikes s–1 until ∼0.4 s, and then increased steadily by 20 spikes s–1
from 0.4 to 1.4 s as the monkey waited for the ‘go’ signal to arrive.
The time course of activity from this neuron was strongly influenced
by the monkey’s knowledge of the random ‘go’ time schedule. Notably,
the responses depicted by the purple and red curves in Figure 3a were
recorded under identical physical conditions: while viewing the same
target and fixation point and awaiting the ‘go’ signal. The marked difference in the time course of the neural response reflects only a difference
in the monkey’s state of anticipation.
The response functions in Figure 3a can be approximated by
a weighted sum of the anticipation functions associated with the
unimodal and bimodal schedules (black curves). The weights derived
from these fits furnish a test of the hypothesis that the LIP response is
dominated by the hazard function associated with the schedule that
the monkey experienced. For the neuron in Figure 3a, the weights
assigned to the bimodal and unimodal anticipation functions (wb and
wu, respectively) were +36.6 ± 1.7 and –8.6 ± 1.1, respectively, during
testing with the bimodal schedule. In contrast, during testing with the
unimodal schedule, the fit was dominated by the unimodal anticipation
function: wb = –10.7 ± 1.3, wu = +8.9 ± 1.3. The negative weights associated with the ‘wrong’ anticipation function suggest that LIP activity
also reflects the knowledge that under each of the schedules certain ‘go’
times will not occur.
These weighting coefficients estimate the change in spike rate
associated with the range of the anticipation functions shown in
Figure 3a (top); they have the unit ‘spikes per second per unit of
anticipation’, where one unit is the range of the subjective hazard
associated with the unimodal distribution (see Methods). The weights
and their standard errors demonstrate that the response modulations
observed in either block are dominated by the appropriate bimodal
or unimodal anticipation function.
Figure 3b shows the averaged responses from all 70 neurons in the two
monkeys. To make these graphs, we normalized the responses from each
neuron to its average firing rate during the delay period using combined
data from both schedules. When the ‘go’ times were drawn from a
bimodal distribution, the response averages showed a triphasic pattern:
increasing in anticipation of ‘go’ times drawn from the early mode, then
decreasing when ‘go’ signals were unlikely to occur, then increasing again
in anticipation of ‘go’ times drawn from the later mode (Fig. 3b, red).
When the ‘go’ times were drawn from the unimodal distribution, the
triphasic pattern was less apparent. The responses showed a steady rise
(monkey J) or remained stable (monkey H) from ∼0.4 s onward.
The graphs demonstrate that anticipation-related modulation can
constitute a large fraction of the delay period activity. For example,
during the waiting period for bimodally distributed 'go' signals, the
responses from monkey J modulated between –40% and +40% of the
delay period activity (Fig. 3b, upper red curve). These effects were
smaller in monkey H, but the effect of schedule on the responses was
highly significant for both monkeys. Fits to the response averages using
weighted sums of the anticipation functions (equation (5); Fig. 3, black
curves) indicate that the schedule affected the pattern of firing in both
monkeys (Table 2). When the schedule changed from unimodal to
bimodal, the fitted response functions showed an increase in the weight
of the bimodal anticipation function on the response and a decrease in
the weight of the unimodal schedule (P < 10–5 for all comparisons in
both monkeys; see Table 2).
Individual neurons showed a variety of response patterns, but most
underwent modulations similar to the response averages and example
in Figure 3a,b. The spike rate functions for 68 of 70 neurons (97%)
were well described by a weighted sum of the two anticipation functions
(P < 0.05; H0: wb = wu = 0 ; mean R2 = 0.67; interquartile range =
0.53–0.87) delayed by 55 ms on average (τ, equation (5)). The scatter
plot in the upper panel of Figure 3c compares the contribution of the
bimodal anticipation function to each neuron’s response under the two
schedules. This bimodal anticipation function was weighted significantly
more during the block using the bimodal schedule of ‘go’ times (mean
difference = +12.2 ± 2.8, P < 0.001, paired t-test). The corresponding
analysis of the unimodal anticipation functions is shown in the lower
panel of Figure 3c. The unimodal anticipation functions tended to exert
less weight overall but were stronger during blocks of trials using the
unimodal schedule of ‘go’ times (mean difference = –5.9 ± 1.3, P < 0.01).
Figure 4 Neural activity is inversely related to reaction time on a trial-bytrial basis. For each trial, we compared the reaction time to the neural
activity in an epoch from 150 ms before to 50 ms after the ‘go’ signal. We
removed the potentially confounding effect of time by subtracting the mean
reaction time and spike rate from each value. The correlation coefficient
was obtained using these residual values. The histogram shows r-values
from 70 cells (shading indicates significance; P < 0.05 in 28/76 neurons;
Fisher z) using all trials from unimodal and bimodal schedules (results
are similar using either schedule alone). The weak but significant inverse
correlation indicates that variability in neural activity in LIP affects the
reaction time.
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Figure 5 Time-dependent anticipatory activity is associated with motor
preparation. (a) Variant of the delayed eye movement task. The monkey
made an eye movement to the green target as soon as the blue stimulus
dimmed. The monkey must attend to both targets but plan an eye
movement only to the green target. Either target could appear in the
neuron’s response field. (b) Single-neuron responses. The average activity
is shown for trials in which the saccade target appeared in the response
field (green) and for trials in which the ‘go’ cue appeared in the response
field (blue). (c) Population responses. Left, response averages from 25
neurons (monkey J) on trials in which the saccade target appeared in the
response field (green) and trials in which the ‘go’ cue appeared in the
response field (blue). Right, population responses of the same 25 neurons
on trials in which both the saccade target and the ‘go’ signal appeared
in the response field (black), along with the average activity of these 25
neurons on trials using the unimodal time schedule with the saccade target
in the response field (purple).
sequences, wb was +33.6, +4.2, and +18.6 spikes s–1 per unit anticipation,
respectively. In unimodal-bimodal-unimodal sequences, wu was +8.3,
+0.1, and +6.9 spikes s–1 per unit anticipation, respectively.
Note that the weights for the appropriate anticipation function tended
to be positive, indicating that greater anticipation leads to higher spike
rates, on average, in LIP.
Together, these findings indicate that persistent activity in LIP is
modulated by the passage of time. All of the neurons in our sample
respond selectively to targets in their response fields. Yet, during the
delay period in which the monkeys knew where but not when to make
an eye movement, LIP neurons modulated their discharge by tens of
spikes per second (on the order of one-third the level of sustained
activity) in a pattern resembling the theoretical anticipation functions
associated with the schedules of random ‘go’ times.
The changes in neural activity associated with the change in task
timing evolved on a short time scale (on the order of minutes) and
were highly flexible. For the neuron in Figure 3a, reliable changes in
the modulation pattern emerged as early as 20 trials after the change
in the time schedule. For example, wb decreased from +36.6 ± 1.7 to
+2.6 ± 1.7 and wu increased from –8.6 ± 1.1 to +22.4 ± 2.0 over the first
20 trials (P < 0.001, data not shown). Notably, this rapid change was also
evident in the monkeys’ behavior. As is apparent from Figure 2a, reaction time tended to be shorter on average under the unimodal schedule
across all ‘go’ times. This change was apparent within 30 trials after the
change from the bimodal to the unimodal time schedule in monkey J
(P < 0.01, t-test).
To further examine this flexibility, we conducted a third block of testing
in 11 neurons (monkey J) in which we reverted to the first schedule of
random ‘go’ times used in the first block (either unimodal or bimodal).
The neural activity in this third block changed again and became similar
to the activity in the first block of trials. In bimodal-unimodal-bimodal
238
Trial-to-trial correlation between neural activity and behavior
Does the anticipatory activity in LIP help the monkey prepare its
eye movement, or are the behavioral and neural manifestations of
anticipation merely coincidental? Although it is impossible to exclude
the latter possibility without manipulating the discharge of the neurons
we recorded, some insight can come from examining the relationship
between neural activity and reaction time on single trials. We asked
whether the firing rate in the epoch around the time of the ‘go’ signal
predicted the monkey’s reaction time on that trial. By subtracting the
mean neural activity and mean reaction time from the data, we removed
the effect of time and anticipation from both sets of measurements,
leaving only the residual changes about the time-dependent means. We
then calculated a correlation coefficient between these residual values
for each neuron in our sample. The histogram of correlation coefficients
(Fig. 4) shows that the majority of neurons showed a negative correlation
between firing rate and reaction time on single trials (median correlation coefficient = –0.09, P < 0.001; for neurons with significant
anticipation modulation, the median correlation coefficient was –0.10,
P < 0.001). The size of this correlation was weak, but it is notable that
the variable discharge of a single neuron on a single trial has any impact
at all on the monkey’s reaction time. Thus, in addition to reflecting the
monkeys’ state of anticipation, the time-dependent activity of LIP neurons is probably partially responsible for the behavioral manifestation
of this anticipation in the saccadic eye movements.
Anticipatory activity is associated with eye movement planning
Previous studies suggest that area LIP plays a role in the allocation
of spatial attention and the planning of eye movements9,26,28. The
anticipatory activity we observed could reflect temporal variation
in either of these functions. To distinguish these possibilities, we
trained the monkeys on a variant of our delayed eye movement task
(Fig. 5a). Instead of anticipating the dimming of the fixation point, the
monkey was required to monitor a second peripheral target, which was
distinguished by its color. The sole purpose of this second target was to
provide the ‘go’ signal, again by dimming slightly, thereby instructing
an eye movement to the other target. Thus the monkey had to attend
to the ‘go’ cue and possibly to the saccade target, but the eye movement
plan was directed only to the latter. By placing either the ‘go’ cue or the
eye movement target in the neuron’s response field, we could determine
whether anticipatory activity in LIP is associated with the direction of
the intended eye movement, spatial attention or both.
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The neuron in Figure 5b showed robust anticipatory activity
associated with the bimodal time schedule when the eye movement
target appeared in the response field (wb = +65.2 ± 3.6, wu = –13.2
± 2.2). In contrast, when the 'go' cue was in the response field, this
anticipatory modulation was much weaker (Fig. 5b; wb = +6.9 ± 1.8,
wu = –13.8 ± 1.0). This pattern of results was similar across the population of 25 neurons tested in this manner (Fig. 5c, left). The average
wb was +24.7 ± 2.5 spikes s–1 per unit anticipation when the target
appeared in the response field compared to +5.1 ± 1.9 when the 'go'
cue appeared in the response field. Although it is not apparent in the
figure, the weaker representation of the bimodal anticipation function
in the latter configuration was significant (P < 0.01, t-test). In contrast,
when these neurons were tested under the unimodal schedule (target
in response field), the responses did not show a positive influence of
the bimodal anticipation function (Fig. 5c, right; mean wb = –4.1 ± 1.2,
mean wu = +6.3 ± 0.9).
To test whether we were suppressing modulation at the attended
location of the 'go' cue because of competition with the saccade target
in the opposite hemifield, we added a control condition in which both
the saccade target and the 'go' cue were in the neuron's response field.
The pattern of neural activity in this condition (Fig. 5c, right) was virtually indistinguishable from the activity when the saccade target was
presented in the RF and the ‘go’ cue outside the response field (green in
Fig. 5c, left). Hence the location of the ‘go’ cue did not seem to influence
the pattern of anticipation responses in LIP.
Based on these control experiments, we conclude that anticipatory
modulation was strongest in LIP neurons that represent the locus of
the intended eye movement.
DISCUSSION
We trained rhesus monkeys to anticipate the timing of a ‘go’ signal in
a delayed saccade task. Unlike in previous studies29, the monkeys had
to learn two probabilistic time schedules that were used in alternating
blocks of trials. To maximize the chances that the animals would learn
both schedules, we used a bimodal and unimodal distribution of ‘go’
times with minimal overlap. Examination of the monkeys’ reaction
times demonstrated that they had learned features of these probabilistic
schedules in order to anticipate the time of the ‘go’ instruction.
We also found that single neurons in area LIP modulated their firing
rate as a function of time in a way that reflected the monkeys’ state of
anticipation. The spike rate was strongly influenced by the probability
that the ‘go’ instruction would occur in the next moment, based on the
monkey’s experience with the two schedules. The spike rate modulation in LIP is thus related to the hazard rates associated with the unimodal and bimodal probabilistic schedules of ‘go’ times (Figs 3 and 5).
The pattern of modulation also reflected the well-known fact that the
experience of elapsed time carries with it a degree of uncertainty that is
proportional to the true duration (Weber’s law)3,5,30. This uncertainty
implies that the distribution of ‘go’ times that the monkey experienced
during training is a distorted version of the probability distributions
that we programmed into the computer, giving rise to the subjective
hazard rates reflected in LIP.
Notably, the temporal pattern of the LIP spike rate often changes substantially within a single experimental session upon a change in the probabilistic schedule. It is important to bear in mind that markedly different
patterns of response (in Fig. 3, for example) were observed from the same
neuron while the monkey viewed the identical display and awaited the
‘go’ instruction. Evidently, the brain can detect the change of schedule
after a few samples of ‘go’ times (<30 trials) and adjusts its circuitry to
achieve the appropriate pattern of anticipation in LIP. Further studies are
needed to clarify the mechanism of this flexibility.
NATURE NEUROSCIENCE VOLUME 8 | NUMBER 2 | FEBRUARY 2005
In the control experiment used to dissociate an intention to move
the eyes from the spatial attention allocated to the detection of the ‘go’
instruction (Fig. 5), we found that the representation of the hazard rate
was markedly diminished when the ‘go’ cue was in the response field
and the saccade target was not. At first glance, this seems to contradict
a recent study that demonstrated hazard-like modulation of attentionrelated signals in area V4 of the monkey29. However, a small amount
of modulation was present when spatial attention was directed to the
LIP response field (∼5 spikes s–1 per unit of anticipation). It is possible
that this is sufficient to explain the modest degree of modulation seen
in the V4 study of attention. Alternatively, the allocation of spatial
attention may be difficult to dissociate from an intention to make an eye
movement31. Indeed, the neural basis of attention-related modulation
in area V4 may be mediated through high-level oculomotor structures32.
It is therefore plausible that the intention-related signals seen in our
experiments could underlie a shift in spatial attention that is not in
competition with an eye movement plan33.
The present study cannot determine whether the timing-related
anticipatory activity arises in area LIP or is simply passed to LIP from
other structures that have been implicated in interval timing, such
as the prefrontal cortex18,20, the basal ganglia34 or the cerebellum35.
However, two observations lead us to suspect that the anticipation
signals measured in LIP may have more than a coincidental role in
shaping the monkey’s behavior. First, the anticipation functions are
reflected by the neurons and by the reaction time at the same latency
with respect to the ‘go’ instruction. The anticipation of the ‘go’ instruction best matches the neural responses with a latency of 55 ms and
is maximally (inversely) correlated with the monkey’s reaction time
56 ms before the ‘go’ signal (Fig. 2).
Second, we observed a weak correlation between the variable
responses on single trials and the monkey’s reaction time (Fig. 4). The
weakness of the reaction time–spike rate correlation distinguishes LIP
from motor structures, such as the frontal eye field and the superior
colliculus, with which it is interconnected36,37. Indeed, much larger
negative correlations between spike rate and reaction time have been
reported in these areas38,39. The weak correlation between the variable
spike rate from LIP neurons on single trials and the monkey’s subsequent
reaction time on that trial suggests that either LIP neurons directly influence reaction time, or they mirror with great fidelity the structures that
lie along the causal chain. This implies that the marked anticipationrelated modulation represented in LIP is likely to explain the behavioral
manifestation of anticipation: reduced saccadic latency.
Our findings extend a previous study that has demonstrated a
possible role for area LIP in interval timing5. As in the present study, LIP
neurons were found to encode the salience of potential eye movement
targets in a dynamic fashion. In that study, the monkey’s perception
of the duration of a test light was explained by comparing the activity
of LIP neurons whose activity increased or decreased as a function of
time. In light of the present findings, we suspect that the modulation
of activity is governed by the anticipation of the termination of the test
light, whose random durations resemble the unimodal schedule of ‘go’
times used in the present study. By adding and subtracting the subjective
hazard rate to or from low and high background firing rates, respectively,
LIP could produce a pair of crisscrossing functions (as in ref. 5) to represent elapsed time with respect to a memorized standard duration.
Together, these studies suggest that LIP encodes elapsed time insofar
as it affects the meaningfulness of visual objects that are potential gaze
targets. Indeed, the observation that some LIP neurons track the motion
of occluded objects40 might be explained by the prediction of a salient
object emerging from behind an occluder in time. In addition to elapsed
time, it has been shown that persistent neural activity in area LIP can
239
© 2005 Nature Publishing Group http://www.nature.com/natureneuroscience
ARTICLES
be influenced by a variety of factors associated with reward expectation12, response bias41 and decision formation42. All of these functions
require the neurons to operate on a time scale governed neither by
immediate changes in the sensory environment nor by real-time constraints of moving body parts. Hence, the brain must keep an internal
representation of time to determine by what time an action must occur
or when information becomes relevant. Throughout the association
cortex4,20,43,44, neurons with persistent activity may therefore represent
elapsed time to infer the temporal structure of the environment.
METHODS
Task. Two rhesus monkeys were trained on the delayed eye movement task45. The
monkeys maintained their gaze within 1° of a fixation spot in the center of the
display. After 400 ms of stable fixation, a red target was presented at a position
between 3° and 15° eccentricity. After a variable delay, the fixation point dimmed
by 30% of its luminance. This dimming served as the 'go' signal to make an eye
movement to the target. A liquid reward was given for accurate saccades (within 4°
of the target) initiated 150–500 ms after the 'go' signal. To encourage fast responses,
reward size was governed by an exponential function of reaction time (minus
150 ms). The time between target onset and the 'go' signal was a random variable
drawn from either a bimodal or a unimodal distribution (Fig. 1b, upper row).
The bimodal distribution, B(t), was the sum of two non-overlapping Rayleigh
distributions, delayed by d1 and d2:
1
B(t) = 2 (R1 ⫹ R2)
where
Ri =
{
2αi(t – di)e –αi(t–di)2
0
for t > di
otherwise
(1)
(α1 = 18, d1 = 0.1, α2 = 15, d2 = 1.75). The ‘go’ time was drawn with equal
probability from R1 or R2. The unimodal distribution of ‘go’ times was a single
Weibull function delayed by 0.5 s:
U(t) =
{
3α(t –
1 2 –α(t– 1 )3
) e
2
2
0
for t >
1
2
otherwise
(2)
The probability distribution of the ‘go’ times was fixed throughout a block of
trials. After 200–300 trials, the time schedule was changed without notification.
Ideally, anticipation should be governed by the conditional probability that an
event will occur given that it has not yet occurred, termed the hazard rate (Fig. 1b,
middle row). This is the probability that the ‘go’ signal will occur at time t divided
by the probability that it has not yet occurred:
h(t) =
ƒ(t)
1 – F(t)
(3)
where f(t) is either U(t) or B(t), and F(t) is the cumulative distribution,
t
∫0 ƒ(s)ds.
To obtain our predicted anticipation functions, we calculated ‘subjective’
hazard rates based on the assumption that elapsed time is known with
uncertainty that scales with time. The probability density function f(t) = U(t)
or B(t) was first blurred by a normal distribution whose standard deviation is
proportional to elapsed time.
1
~
ƒ(t) =
φt
∞
∫ƒ(t)e –(τ–t)2/(2φ2t2)dτ
2π –∞
(4)
The coefficient of variation, φ, is a Weber fraction for time estimation (for
most analyses, φ = 0.26). Equation (4) implements the idea that the monkey’s
estimate of elapsed time carries uncertainty. Thus, an event at objective time t0
is sensed as if it occurred at t0 ± σ. The subjective hazard rate is then obtained by
240
~
~
substituting f(t) and its definite integral, F(t) into equation (3). We refer to these
subjective hazard rates as anticipation functions, Au(t) and Ab(t), below. For ease
of interpretation, they are scaled in all graphs and fits by a common factor (2.03,
for φ = 0.26) so that the functions range from 0 to 1 (Fig. 1b, bottom row).
In the cue experiment, a blue and a green stimulus were presented on opposite
sides of the fixation point. The monkeys' task was to make a saccade to the green
stimulus (the target) as soon as the blue stimulus (the cue) dimmed. We compared the activity during trials in which the ‘go’ cue was in the neuron's response
field and the target was outside the response field, to trials in which the opposite
configuration was present. To determine any potential color selectivity46 we also
recorded trials in which the monkey made saccades to either a green or a blue
target in interleaved blocks of trials.
Recording procedure. Action potentials from single neurons were recorded extracellularly using standard procedures10 and stored with 1 ms precision. Eye position was monitored using a scleral search coil (sampled at 1 kHz). We screened
neurons using a memory-guided saccade task45. Only neurons with high spatially
selective delay period activity were studied: the average delay activity (computed
320–720 ms after target onset) was at least 60% of the visual response (80–320 ms
after target onset). Most of these neurons also showed strong presaccadic activity
(88% had larger responses in a 100 ms epoch ending at saccade initiation than in
the 300 ms epoch preceding this; P < 0.05, t-test). This property is typical of LIP
neurons in other studies10,24,37,47. All training, surgical and recording procedures
complied with the guidelines of the National Institutes of Health and the research
protocols approved by the University of Washington.
Data analysis. For each trial, the spike rates were truncated at 50 ms after the
dimming of the fixation point. The first 50 trials after the change in the time
schedule were excluded from the analysis. We fitted the mean spike rates (80-ms
bins) after the initial visual response (starting at 160 ms after target onset) using
a weighted sum of subjective hazard rates:
r(t) = we + wuAu(t – τ) + wbAb(t – τ) + ε
(5)
where r is the neuronal response, we is a constant term, and wu and wb are the
weights for the unimodal (Au) and bimodal (Ab) anticipation functions, respectively, delayed by time shift τ. ε represents noise, which is assumed to be Gaussian
with uncertainty derived from the sample means. Equation (5) was also used
to fit the reaction times on each trial with a weighted sum of subjective hazard
rates. Because the functions range from 0 to 1, the weights can be interpreted
as the magnitude of the spike rate modulation attributed to these theoretical
waveforms (an approximation to a basis set) in units of spikes per second per
unit anticipation (for the fits to the reaction times, the units are milliseconds
per unit anticipation).
We used a maximum-likelihood fitting procedure to obtain the fits, parameter
estimates and their standard errors. Standard errors of parameters were estimated
from the Hessian matrix of second partial derivatives of the log likelihood48 and
were used to generate t-statistics cited throughout the paper. We fit the data from
each neuron and each schedule independently to obtain the weights shown in the
scatter plots (Fig. 3c). The Weber fraction was fixed (equation (4), φ = 0.26). We
tried to estimate φ using the fits, but found that only strongly modulated neurons
gave reliable estimates (mean φ from 48 reliable cases was 0.33 ± 0.03). A similar
strategy was used to fit the population response data (Fig. 3b) except that fits to
the two response averages were constrained to use a common time delay, τ, and
the Weber fraction was free. The large Weber fraction obtained from these fits
(0.41 and 0.5 for monkeys J and H, respectively) was probably induced by smearing of the response functions because of averaging. For the all other analyses, we
assumed φ = 0.26, consistent with previous human and animal studies30,49 and
with behavioral measurements in monkeys in our laboratory5.
The partial correlation coefficients between the theoretical anticipation function
and the reaction times (RT) observed under either the unimodal or bimodal schedule were calculated by partitioning the 3 × 3 covariance matrix based on the ordered
triplets [RT(t), Au(t – τ), Ab(t – τ)], where t is the ‘go’ time and τ = 56 ms (from the
fit shown in Fig. 2). The partitioning effectively factors out the contribution of the
potentially confounding variable (for example, Au(t – τ) for data obtained with the
bimodal schedule) on the correlation between the other two variables50.
VOLUME 8 | NUMBER 2 | FEBRUARY 2005 NATURE NEUROSCIENCE
ARTICLES
© 2005 Nature Publishing Group http://www.nature.com/natureneuroscience
For the spatial control experiment (Fig. 5), we also tested for color selectivity by
interleaving blocks of trials in which the monkey made saccades to either a green
or a blue target. Only 2 of 18 neurons responded significantly more to one color.
The mean absolute response difference between the two colors was 3.6 spikes s–1
(s.d. = 4.0, which is >0.75 times the mean, consistent with the expected distribution of absolute values under hypothesis H0: no difference).
ACKNOWLEDGMENTS
We thank M. Mihali and L. Jasinski for technical assistance, and T. Yang,
T. Hanks, M. Leon and J. Palmer for helpful comments on the manuscript. Work was
supported by Howard Hughes Medical Institute, the International Human Frontiers
Science Program Organization, the Fonds voor Wetenschappelijk Onderzoek
Vlaanderen, the National Center for Research Resources (RR00166) and the
National Eye Institute (EY11378).
COMPETING INTERESTS STATEMENT
The authors declare that they have no competing financial interests.
Received 17 September; accepted 20 December 2004
Published online at http://www.nature.com/natureneuroscience/
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